Mathematical Modeling of an Economic Cycle Model with Harmonic and Random Drivers

Applied mathematics has been widely applied in various fields of science and engineering. Among these applications, the study of economic cycles is a very important branch. It is because that an economic system usually develops in a repetitive manner of expansion and contraction under many deterministic and random factors. It exhibits a complex phenomenon of nonlinearity and randomness. In this paper, a nonlinear economic cycle model is proposed to consider government adjustment and market uncertainty. The periodic income adjustment from government is applied to the model as a harmonic function with income. Meanwhile, Gaussian white noise is employed to model market uncertainty. The response of the nonlinear economic cycle model is investigated on its probability density function (PDF) using a path integral method. The short-time Gaussian approximation technique is introduced to evaluated the transition PDF between the two successive time intervals, and the Gauss-Legendre integration scheme is applied to the path integration. Two cases are considered in the following analysis. One is about the high-level government adjustment. The other is about the high-level market uncertainty. The results show that the joint PDF of income and income growth rate will reach a stable state and it periodically evolve along with time.